模簇的Chern类
Chern classes of modular varieties
摘要：
设X是Hermitian局部对称空间。我们证明了X的每个Chern类对于X的B-Borel-Satake紧化X* 的上同调都有一个标准提升，并且由此得到的Chern数对于X的紧对偶X^满足Hirzebruch比例公式.对于X上的任何自守向量丛代替切丛，同样的结果也成立。因此，存在由这些提升类生成的H*（X*）的子代数到H*（X^）上的满射。证明的方法是在X* 的奇异层附近的这些丛上构造纤维状平坦联络，然后在那里找到从层中拉回的陈类的de Rham代表。
英文摘要：
Let X be a Hermitian locally symmetric space. We prove that every Chern class of X has a canonical lift to the cohomology of the Baily- Borel-Satake compactification X* of X and that the resulting Chern numbers satisfy the Hirzebruch proportionality formula with respect to the compact dual X^ of X. The same result holds for any automorphic vector bundle over X in place of the tangent bundle. As a consequence there is a surjection of the subalgebra of H*(X*) generated by these lifted classes onto H*(X^). The method of proof is to construct fiberwise flat connections on these bundles near the singular strata of X*, where one then finds de Rham representatives of the Chern classes which are pulled back from the strata.
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